If it were cheap enough, maybe, but at the moment the demand fluctuations are covered by power generating plants which come online in times of high demand (e.g. day) and shut off during low demand (e.g. night). Typically these plants burn natural gas.
Sufficiently cheap storage would be very useful, yes.
Actually, pumped storage hydro is used for the purposes than DanArmak describes; see my post elsewhere in this thread.
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.